Week 4 Quantum

Question 1

What does {Ψ} represent?

Answer 1

A basis

Question 2

What is a Wavefunction expansion?

Answer 2

A Wavefunction expressed as a linear sum of eigenfunctions

Question 3

What does Ψ (x) = Sum of Cn Ψn (x) mean?

Answer 3

A wavefunction expression - expressing a wavefunction as a linear sum of eigenfunctions

Question 4

What is wavefunction orthonormality?

Answer 4

Eigenfunctions are orthogonal t each other and magnitude of [Ψn (x) ]^2 =1

Question 5

Express orthonormality symbolically

Answer 5

∫ Ψn* (x) Ψm (x) dx = 1 if n=m and 0 otherwise (from infinity to - infinity)

Question 6

What is the expansion coefficient?

Answer 6

Cn

Question 7

How to calculate expansion coefficient?

Answer 7

C n = ∫ Ψn* (x) Ψm (x) dx (infinity to -infinity)

Question 8

What form can you write any vector in?

Answer 8

The sun of basis-vectors (basis expansion)

Question 9

What is the concept behind matrix mechanics?

Answer 9

Wavefunctions are vector that live in Hilbert Space and the eigenvectors are the basis-vectors of tha space

Question 10

What is aHilbert Space?

Answer 10

An infinite-dimensional linear complex vector space with well-defined properties

Question 11

What is another term for eigenfunctions?

Answer 11

Eigenvectors

Question 12

What is a basis in 3D wavefunction notation?

Answer 12

Ψ = ∑Cn Ψ n
- Summation, n=1 below, 3above

Question 13

What is ket?

Answer 13

State vector in Hilbert space
- column vector

Ψ>

Question 14

What is bra?

Answer 14

Conjugate transpose of ket
- row vector

<Ψ|

Question 15

Write ket in wave and Dirac

Answer 15

|Ψ> = Ψ (x)

Question 16

Write eigen vector in Dirac and wave notation

Answer 16

Ψ > = Ψn (x)

Question 17

What does | En> = | Ψ> mean?

Answer 17

The eigenvector labelled with its eigenvalue
- the state with the En eigenvalue

Question 18

What is | Ψ>?

Answer 18

The eigenvector

Question 19

What does dagger mean?

Answer 19

That it is the Hermitian adjoint of the matrix

Question 20

How do you get the Hermitian adjoint of a matrix

Answer 20

Transpose (flip rows with columns), then complex conjugate

Question 21

What is the inner product?

Answer 21

Takes 2 vectors and computes the sum to their product to return a scalar

Question 22

What is the inner product in Dirac and wavefunction

Answer 22

<Ψn| Ψm> = ∫ Ψn* (x) Ψm (x) dx

• infinity to -infinity

Question 23

Write bra in wave and Dirac

Answer 23

<Ψ| = Ψ*

Question 24

What is ket in vector notation?

Answer 24

(A0
A1
A2..

Question 25

What is bra in vector notation?

Answer 25

A0* A1* A2*…

Question 26

Write the Wavefunction expansion in Dirac and vector

Answer 26

| Ψ> = ∑ Cn |n> = (c0 C1 C2

Question 27

Write orthonormality in Dirac and vector notation

Answer 27

<Ψn| Ψm > = δnm Ei.ej = δij

Question 28

How are bra and ket related?

Answer 28

Conjugate transpose (dagger)

Question 29

How to write operators in Dirac , wavefunction

Answer 29

Q hat | Ψ> = qn | Ψ> Q hat Ψn (x) = qn Ψn (x)

Question 30

How to determine if Hermitian?

Answer 30

Check whether equal to conjugate transpose Real is symmetric, complex = its own conjugate transpose

Question 31

How to determine if a wave is an eigenstate of an operator?

Answer 31

Show that Q hat | Ψ> = qn | Ψ> - use operator and see if same result

Question 32

Significance of Hermitian operators

Answer 32

- Correspond to physical quantities - Real eigenvalues representing the possible measurement outcomes

Question 33

Mathematical properties of Hermitian operators

Answer 33

Orthogonal eigenvectors and diagonalizable

Question 34

What does an operator do?

Answer 34

Acts on wavevector, no meaning in isolation

Question 35

What is the result of an operator acting on a wavector?

Answer 35

Another wavevector

Question 36

Depict operator acting on a wavefunction in Dirac

Answer 36

Q | Ψ> = |Q Ψ> OR <Ψn | Q† = < Q† Ψ | Where Q = Q hat

Question 37

What does

Answer 37

Q hat † Ψn* (x)

Question 38

Write condition for Hermitian operators in Dirac

Answer 38

<Ψn | QΨm> = Where Q = Qhat

Question 39

How to prove momentum operator is Hermitian

Answer 39

- Apply operator - Integrate by parts U = state vector Dv = Complex wave/dx - Uv - ∫vdu -> 1st terms cancels because wave goes to zero at limits

Question 40

Write expectation value in dirac

Answer 40

= <Ψ |Q| Ψ>

Question 41

How else can you write <Ψ |Q| Ψ>?

Answer 41

<Ψ |Q Ψ>

Question 42

What is the concept behind quantum operators as matrices?

Answer 42

An operator acting on a vector is also a vector as with wavefunctions - operator like an mxm matrix

Question 43

What is the probability of measuring the eigenvalue En?

Answer 43

Cn^2

Question 44

Write an operator as a matrix in Dirac

Answer 44

Q ij = <Ψi | Q | Ψj> The ij-th element in row i and column j

Question 45

How to express Hamiltonian as a matrix?

Answer 45

- ij-th element is <Ψi | H | Ψj> = Ej <Ψi | Ψj> = Ej δij via orthonormality - where H | Ψj> = Ej| Ψj> - H = hbar w [1/2 0 0 [ 0 3/2 0 [ 0 0 5/2

Question 46

How do we express eigenstates as eigenvectors?

Answer 46

As orthonormality basis vectors - | Ψo> = 1 | Ψ1> = 0 | Ψ2> = 0 0 1 0 0 0 1 -H | Ψo> = 1* (mxm matrix) | Ψ1> = 0 * (mxm matrix) | Ψ2> = 0 0 1 0 0 0 1

Question 47

What is pairwise multiplication?

Answer 47

Operation of multiplying corresponding elements from 2 vectors of same length E.g A * B = C where Ci = Ai * Bi

Question 48

What conditions for Hermitian conjugate of an operator?

Answer 48

The operator that satisfies: - <Ψn | Q Ψm> = Meaning Q operating on n wave then taking inner product with m is that same as Q † on m wave followed by inner product with n

Question 49

What are the 6 different representations of Hermitian conjugate operators in Dirac?

Answer 49

(Q + R) † = Q † + R † ,(aQ) † = a*Q † (Q | Ψ> ) † = <Ψ|Q † (Q | Ψ> †) = | Q Ψ> † = = <Ψ | Q Ψ> = <Ψ | Q| Ψ> | Q Ψ> † = <Ψ | Q Q= Q †

Question 50

What are these {}?

Answer 50

Poisson brackets - commutator

Question 51

What is the commutation relation formula?

Answer 51

[A , B ] = AB - BA

Question 52

What is the DeBroglie hypothesis in commutators

Answer 52

[ x, p ] = x p - px. = ih

Question 53

What are the canonical commutation relations?

Answer 53

The fundamental relations governing the behaviour of position and momentum operators

Question 54

What are formulas for the canonical commutation relations?

Answer 54

[ x, p ] = ih {x, p } = 1

Question 55

What is the product of 2 operators?

Answer 55

An operator

Question 56

What is the physical interpretation of commutation?

Answer 56

Order of measurements not important - can measure both simultaneously

Question 57

What is formula if operators do commute in dirac?

Answer 57

[ G, Q] | Ψ> = 0

Question 58

How to assess position/momentum commutation?

Answer 58

[ x , p ] | Ψ> = x p | Ψ> - p x | Ψ> - will need to differentiate by parts

Question 59

What is the uncertainty relation in words?

Answer 59

Tells you the precision you can obtain if try to measure 2 quantities

Question 60

How to calculate uncertainty relation?

Answer 60

Δg Δ q >/= 1/2 [ < [ G, Q] > ]

Question 61

How to derive Heisenberg’s uncertainty relation?

Answer 61

Δx Δp >/= 1/2 [ < [x , p ] > ] >/= 1/2

Question 62

What happens if don’t commute?

Answer 62

A Psi n is an eigenstate of B with eigenvalue Bn - They have a shared eigenbasis

Question 63

Formula for anti-commutator

Answer 63

{A,B} = AB + BA